Abstract
The integer sequence defined by Pn+3 = Pn+1 + P-n with initial conditions P-0 = 1 and P-1 = P-2 = 0 is known as the Padovan sequence {P-n}(n is an element of Z). A recurrence sequence {u(n)}(n is an element of Z) is said to be of Padovan-type if it satisfies the same recurrence relation as the Padovan sequence but with arbitrary initial values u(0), u(1), u(2) not all zero. The most famous Padovan-type sequence is the Perrin sequence given by u(0) = 3, u(1) = 0 and u(2) = 2. We show that every Padovan-type sequence has at most 2 zeros, except for nonzero multiples of shifts of the Padovan sequence which has 0-multiplicity 5. We also show that {P-n}(n is an element of Z )has total multiplicity 62, i.e., there are 62 pairs (m, n) is an element of Z(2) with m < n for which P-m = P-n. As a consequence, we found that {P-n}(n is an element of Z) has multiplicity 9, being 1 the most repeated Padovan number. Finally, we prove that the Perrin sequence has exactly 1 zero, total multiplicity 23 and multiplicity 4.